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We consider the sedimentation of $N$ spherical particles with identical radii$R$ in a Stokes flow in $\mathbb R^3$. The particles satisfy a no-slip boundarycondition and are subject to constant gravity. The dynamics of the particles ismodeled by Newton's law but with very small particle inertia as $N$ tends toinfinity and $R$ to $0$. In a mean-field scaling we show that the evolution ofthe $N$-particle system is well approximated by the Vlasov-Stokes equation. Incontrast to the transport-Stokes equation considered in the first part of thisseries, \cite{HoferSchubert23}, the Vlasov-Stokes equation takes into accountthe (small) inertia. Therefore we obtain improved error estimates. We alsoimprove previous results on the Cauchy problem for the Vlasov-Stokes equationand on its convergence to the transport-Stokes equation in the limit ofvanishing inertia. The proofs are based on relative energy estimates. Inparticular, we show new stability estimates for the Vlasov-Stokes equation inthe $2$-Wasserstein distance. By combining a Lagrangian approach with a studyof the energy dissipation, we obtain uniform stability estimates for arbitrarysmall particle inertia. We show that a corresponding stability estimatecontinues to hold for the empirical particle density which formally solves theVlasov-Stokes equation up to an error. To this end we exploit certain uniformcontrol on the particle configuration thanks to results in the first part\cite{HoferSchubert23}.

Sedimentation of particles with very small inertia II: Derivation, Cauchy problem and hydrodynamic limit of the Vlasov-Stokes equation